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# zero elements

Let $S$ be a semigroup. An element $z$ is called a *right zero* [resp. *left zero*] if $xz=z$ [resp. $zx=z$] for all $x\in S$.

An element which is both a left and a right zero is called a *zero element*.

A semigroup may have many left zeros or right zeros, but if it has at least one of each, then they are necessarily equal, giving a unique (two-sided) zero element.

More generally, these definitions and statements are valid for a groupoid.

It is customary to use the symbol $\theta$ for the zero element of a semigroup.

###### Proposition 1.

If a groupoid has a left zero $0_{L}$ and a right zero $0_{R}$, then $0_{L}=0_{R}$.

###### Proof.

$0_{L}=0_{L}0_{R}=0_{R}$. ∎

###### Proposition 2.

If $0$ is a left zero in a semigroup $S$, then so is $x0$ for every $x\in S$.

###### Proof.

For any $y\in S$, $(x0)y=x(0y)=x0$. As a result, $x0$ is a left zero of $S$. ∎

###### Proposition 3.

If $0$ is the unique left zero in a semigroup $S$, then it is also the zero element.

###### Proof.

By assumption and the previous proposition, $x0$ is a left zero for every $x\in S$. But $0$ is the unique left zero in $S$, we must have $x0=0$, which means that $0$ is a right zero element, and hence a zero element by the first proposition. ∎

## Mathematics Subject Classification

20N02*no label found*20M99

*no label found*

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